This post will give an intuitive interpretation of the presence of the combination formula (which equals the binomial coefficient) in math problem solutions and probability distributions that are seemingly unrelated to combinations. For example, the binomial coefficient shows up in the probability mass function of the binomial distribution and the negative binomial distribution.
Wikipedia has the following definition for combination:
In mathematics a combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruit, an apple, orange and pear say, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally a k–combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient
The binomial coefficient indexed by n and k is denoted 
is![\left[\begin{array}{ccccc} 1 & X_{1}^{(1)} & X_{2}^{(1)} & \cdots & X_{n}^{(1)}\\ 1 & X_{1}^{(2)} & X_{2}^{(2)} & \cdots & X_{n}^{(2)}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & X_{1}^{(m)} & X_{2}^{(m)} & \cdots & X_{n}^{(m)}\end{array}\right]](https://s0.wp.com/latex.php?latex=%5Cleft%5B%5Cbegin%7Barray%7D%7Bccccc%7D+1+%26+X_%7B1%7D%5E%7B%281%29%7D+%26+X_%7B2%7D%5E%7B%281%29%7D+%26+%5Ccdots+%26+X_%7Bn%7D%5E%7B%281%29%7D%5C%5C+1+%26+X_%7B1%7D%5E%7B%282%29%7D+%26+X_%7B2%7D%5E%7B%282%29%7D+%26+%5Ccdots+%26+X_%7Bn%7D%5E%7B%282%29%7D%5C%5C+%5Cvdots+%26+%5Cvdots+%26+%5Cvdots+%26+%5Cddots+%26+%5Cvdots%5C%5C+1+%26+X_%7B1%7D%5E%7B%28m%29%7D+%26+X_%7B2%7D%5E%7B%28m%29%7D+%26+%5Ccdots+%26+X_%7Bn%7D%5E%7B%28m%29%7D%5Cend%7Barray%7D%5Cright%5D&bg=ffffff&fg=000&s=1&c=20201002)
and the vector containing the Y variable’s values is labeled 
, then the OLS estimation for the coefficients can be calculated by solving 
for 